Starting with a pot of cold tap water, I want to cook a hard-boiled egg using the minimum amount of energy. Is it more energy efficient to bring a pot to boil first and then put the egg in it, or to put the egg in the pot of cold water first and let it heat up with the water?

Green wisdom says: 1. Put eggs in stove, pour water 2. Wait for water to boil 3. Switch off stove 4. Wait some minutes (browse this site) 5. Enjoy eggs. As an aside, I do not support such questions on this site. Atleast until the physicstheory.SE starts. Hence the downvote. I do not see any point in even doing back of the envelop calculations for this.
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ApproximistMar 19 '11 at 4:03

If I +1 this I'm not going to admit it. How about: Expose egg to near vacuum which exists in a very high percentage of the volume of the universe (0 Joules).
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Carl BrannenMar 19 '11 at 7:36

RE phase diagram: Eggs "boil" as water does, there is water and rather high molecular proteins in an egg, which raise boiling point marginally only. Difference is the tendency to overheat, due to lack of "seed" and viscosity.
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GeorgMar 19 '11 at 11:24

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@Carl - the cost if the ticket is rather high
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Martin BeckettMar 19 '11 at 14:30

@Martin Beckett; The OP didn't ask what the most efficient way to boil an egg is "on the earth". So one must assume that he means at a randomly chosen spot. Which is therefore likely to be intergalactic space.
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Carl BrannenMar 20 '11 at 3:46

+1 The reason cookbooks don't recommend this method is it takes different amounts of time for water to start boiling, depending on both the range and the amount of water. So if you don't use the same range and the same pot (with the same amount of water) all the time, the egg won't come out consistently the way you want it.
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Peter Shor Mar 23 '11 at 14:49

"Energy efficient" usually means "with minimal production of entropy". For that you need, 1) a well insulated (on the sides) pot, 2) minimal loss from heating element, 3) the smallest possible (and practical) amount of water amount that can boil an egg (i.e. conver its whole surface) and 4) put the egg at the beginning of the process (not drop it once the water boils because that generates extra entropy because of the temperature gradient).

My cookbook says an egg should simmer in water for about 10 minutes. If you wait till the water simmers and then add the egg, the water will start boiling again in a few seconds, then needs to simmer for the 10 minutes. The heat of vaporization loss takes place for about 10 minutes.
If the egg is placed in cold water first, it'll cook in fewer than 10 minutes of simmering as the pot heats up to boiling, so the heat of vaporization loss will be less than 10 minutes.

A cooked egg has a lower free energy at room temperature and pressure than an uncooked egg (see this link, protein denaturation is complicated, but generally exothermic at room temperature). So boiling an egg does not require energy, it releases energy.

So it is most energy efficient to put the egg and water in a closed container, and use a reversible process to heat the water to boiling temperature. Then use the same pump in reverse to remove the heat. This cycle will net you a little bit of energy when all is said and done.

One simple way to do this is have the water touching a gas through a metal wall in an insulated container, so that the heat can flow, but the wall can sustain a large pressure difference. Then compress the gas slowly to raise the temperature of the gas and water to boiling temperature (this requires an enormous room-sized air-vapor container). If you do it slowly enough, the egg will be cooked when you reach 100 degrees, then you slowly relax the piston back to its starting position, recovering all the energy you put in, and a little more, due to the heat released by the egg as it was cooking.

This is a theoretical answer. The right answer for green purposes is that residental energy consumption is not a significant enough fraction of the total energy consumption for these measures to have any impact. So this type of conservation doesn't do anything, except keep yourself and others aware. To do something real, drive less and encourage energy efficient processes at work.